How to prove the following identity

amitech

1. Homework Statement

e_j=g_(jk)e^k

where e_j is a covariant vector base
e^k is a a contravariant vector base
g_(jk) is the covariant metric

2. Homework Equations

3. The Attempt at a Solution

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HallsofIvy

Homework Helper
What is the definition of the "covariant metric". How do you go from a covariant to the corresponding contravariant vector and vice-versa?

amitech

Hi, by saying "covariant metic" I mean writing the metric in its covariant form - basically g(contra)=[g(cov)]^(-1)
you can use the metric to lower/raise an index on a vector component but here e_j and e^k are not components but the actual unit vectors.

amitech

Hi, by saying "covariant metic" I mean writing the metric in its covariant form - basically g(contra)=[g(cov)]^(-1)
you can use the metric to lower/raise an index on a vector component but here e_j and e^k are not components but the actual unit vectors.

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